Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,16)(8,15)(9,13)(10,12)(11,14)(17,20)(18,19);(5,7)(6,9)(8,13)(10,19)(11,17)(12,16)(14,18)(15,20);(2,4)(5,8,11,10)(6,12,14,15)(7,13,17,19)(9,16,18,20);(5,9,11,18)(6,13,14,19)(7,12,17,15)(8,16,10,20);(6,14)(7,17)(12,15)(13,19)Polycyclic presentation:
2^8|c^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=df,e^b=eh,f^b=fh,f^d=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6993881, #6993884, #6993895, … |
| maximal normal subgroup of | #6993881, #6993884, #6993895, … |
| maximal quotient of | #6358848, #6358856, #6358864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54065);